Theorem nfraldw 3222

Description: Deduction version of nfralw 3224. Version of nfrald 3223 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.)

Hypotheses

Ref	Expression
nfraldw.1	⊢ Ⅎ𝑦𝜑
nfraldw.2	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfraldw.3	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfraldw	⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfraldw

Step	Hyp	Ref	Expression
1		df-ral 3142	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
2		nfraldw.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcvd 2977	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦)
4		nfraldw.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5	3, 4	nfeld 2988	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
6		nfraldw.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
7	5, 6	nfimd 1894	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓))
8	2, 7	nfald 2346	. 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
9	1, 8	nfxfrd 1853	1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4  ∀wal 1534  Ⅎwnf 1783   ∈ wcel 2113  Ⅎwnfc 2960  ∀wral 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142

This theorem is referenced by:  nfralw  3224  nfrexd  3306